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<h1 id="勒让德实习报告">勒让德实习报告 </h1>
<br>
<p>常用的递推算法主要有４种：<code>标准向前列递推算法</code>、<code>标准向前行递推算法</code>、<code>跨阶次递推算法</code>和<code>Ｂｅｌｉｋｏｖ递推算法</code>。<br>
本次采用标准向前行递推法<br>
<br></p>
<h2 id="标准向前行递推法">标准向前行递推法 </h2>
<h3 id="1-算法原理">1. 算法原理 </h3>
<blockquote>
<p>算法流程<br>
<img src="liucheng.jpg" alt="img"></p>
</blockquote>
<ol>
<li>
<p>首先求出种子点数<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>P</mi><mo>ˉ</mo></mover><mrow><mn>0</mn><mo separator="true">,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\bar{P}_{0,0}(\sin\varphi)=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1062em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span><br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>P</mi><mo>ˉ</mo></mover><mrow><mn>1</mn><mo separator="true">,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mn>3</mn></msqrt><mi>sin</mi><mo>⁡</mo><mi>φ</mi></mrow><annotation encoding="application/x-tex">\bar{P}_{1,0}(\sin\varphi)=\sqrt{3}\sin\varphi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1062em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1505em;vertical-align:-0.1944em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9561em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">3</span></span></span><span style="top:-2.9161em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
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c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0839em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span></span></span></span></span><br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>P</mi><mo>ˉ</mo></mover><mrow><mn>1</mn><mo separator="true">,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mn>3</mn><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>sin</mi><mo>⁡</mo></mrow><mn>2</mn></msup><mi>φ</mi><mo stretchy="false">)</mo></mrow></msqrt></mrow><annotation encoding="application/x-tex">\bar{P}_{1,1}(\sin\varphi)=\sqrt{3(1-\sin^2\varphi)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1062em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.84em;vertical-align:-0.4952em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3448em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord">3</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mop"><span class="mop">sin</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8719em;"><span style="top:-3.1208em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose">)</span></span></span><span style="top:-3.3048em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90
l0 -0
c4,-6.7,10,-10,18,-10 H400000v40
H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7
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c53.7,-170.3,84.5,-266.8,92.5,-289.5z
M1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4952em;"><span></span></span></span></span></span></span></span></span></span></p>
</li>
<li>
<p>求出每一行最后两个数的值<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>P</mi><mo>ˉ</mo></mover><mrow><mi>l</mi><mo separator="true">,</mo><mi>l</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mn>2</mn><mi>l</mi><mo>+</mo><mn>1</mn></mrow></msqrt><mi>sin</mi><mo>⁡</mo><mi>φ</mi><msub><mover accent="true"><mi>P</mi><mo>ˉ</mo></mover><mrow><mi>l</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mi>l</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>l</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\bar{P}_{l,l-1}(\sin\varphi)=\sqrt{2l+1}\sin\varphi\bar{P}_{l-1,l-1}(\sin\varphi),l\ge 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1062em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2255em;vertical-align:-0.2861em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9394em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-2.8994em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
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c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1006em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span><br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>P</mi><mo>ˉ</mo></mover><mrow><mi>l</mi><mo separator="true">,</mo><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mfrac><mrow><mn>2</mn><mi>l</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>l</mi></mrow></mfrac></msqrt><mi>cos</mi><mo>⁡</mo><mi>φ</mi><msub><mover accent="true"><mi>P</mi><mo>ˉ</mo></mover><mrow><mi>l</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mi>l</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>l</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\bar{P}_{l,l}(\sin\varphi)=\sqrt{\frac{2l+1}{2l}}\cos\varphi\bar{P}_{l-1,l-1}(\sin\varphi),l\ge 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1062em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.44em;vertical-align:-0.7634em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6766em;"><span class="svg-align" style="top:-4.4em;"><span class="pstrut" style="height:4.4em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.6366em;"><span class="pstrut" style="height:4.4em;"></span><span class="hide-tail" style="min-width:1.02em;height:2.48em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="2.48em" viewBox="0 0 400000 2592" preserveAspectRatio="xMinYMin slice"><path d="M424,2478
c-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514
c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20
s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121
s209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081
l0 -0c4,-6.7,10,-10,18,-10 H400000
v40H1014.6
s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185
c-2,6,-10,9,-24,9
c-8,0,-12,-0.7,-12,-2z M1001 80
h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7634em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span></p>
</li>
<li>
<p>求出每一列的前面的数<br>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>P</mi><mo>ˉ</mo></mover><mrow><mi>l</mi><mo separator="true">,</mo><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mfrac><mrow><mn>4</mn><msup><mi>l</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>l</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow></mfrac></msqrt><mi>sin</mi><mo>⁡</mo><mi>φ</mi><msub><mover accent="true"><mi>P</mi><mo>ˉ</mo></mover><mrow><mi>l</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo stretchy="false">)</mo><mo>−</mo><msqrt><mrow><mfrac><mrow><mn>2</mn><mi>l</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>l</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mfrac><mrow><mo stretchy="false">(</mo><mi>l</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow><mrow><msup><mi>l</mi><mn>2</mn></msup><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup></mrow></mfrac></mrow></msqrt><msub><mover accent="true"><mi>P</mi><mo>ˉ</mo></mover><mrow><mi>l</mi><mo>−</mo><mn>2</mn><mo separator="true">,</mo><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>l</mi><mo>≥</mo><mn>2</mn><mtext>且</mtext><mi>m</mi><mo>≤</mo><mi>l</mi><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\bar{P}_{l,m}(\sin\varphi)=\sqrt{\frac{4l^2-1}{l^2-m^2}}\sin\varphi \bar{P}_{l-1,m}(\sin\varphi)-\sqrt{\frac{2l+1}{2l-3}\frac{(l-1)^2-m^2}{l^2-m^2}} \bar{P}_{l-2,m}(\sin\varphi) ,l\ge 2且m\le l-2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1062em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.44em;vertical-align:-0.7822em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6578em;"><span class="svg-align" style="top:-4.4em;"><span class="pstrut" style="height:4.4em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4171em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.6178em;"><span class="pstrut" style="height:4.4em;"></span><span class="hide-tail" style="min-width:1.02em;height:2.48em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="2.48em" viewBox="0 0 400000 2592" preserveAspectRatio="xMinYMin slice"><path d="M424,2478
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h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7822em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">φ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.44em;vertical-align:-0.7773em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6627em;"><span class="svg-align" style="top:-4.4em;"><span class="pstrut" style="height:4.4em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">3</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.6227em;"><span class="pstrut" style="height:4.4em;"></span><span class="hide-tail" style="min-width:1.02em;height:2.48em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="2.48em" viewBox="0 0 400000 2592" preserveAspectRatio="xMinYMin slice"><path d="M424,2478
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</li>
</ol>
<br>
<h3 id="2-代码解释">2. 代码解释 </h3>
<blockquote>
<p>流程图</p>
</blockquote>
<div class="mermaid">flowchart LR
    A[开始] --&gt; B[计算前&lt;br&gt;三个值]
    B--&gt;|推导|C[得到每一行&lt;br&gt;最后两个数]
    C==&gt;D[每一行&lt;br&gt;前面的数]

</div><blockquote>
<p>向前递推法核心算法</p>
</blockquote>
<pre data-role="codeBlock" data-info="python" class="language-python python"><code><span class="token comment">#正规化勒让德系数</span>
<span class="token keyword keyword-def">def</span> <span class="token function">jisuan_P</span><span class="token punctuation">(</span>n<span class="token punctuation">,</span> sita_rad<span class="token punctuation">)</span><span class="token punctuation">:</span>
    <span class="token triple-quoted-string string">"""计算Plm数组的值，并返回该数组"""</span>
    Plm <span class="token operator">=</span> <span class="token builtin">list</span><span class="token punctuation">(</span><span class="token punctuation">)</span>
    <span class="token keyword keyword-for">for</span> row <span class="token keyword keyword-in">in</span> <span class="token builtin">range</span><span class="token punctuation">(</span>n<span class="token operator">+</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">:</span>
        Plm<span class="token punctuation">.</span>append<span class="token punctuation">(</span><span class="token punctuation">[</span><span class="token punctuation">]</span><span class="token punctuation">)</span>
        <span class="token keyword keyword-for">for</span> clumn <span class="token keyword keyword-in">in</span> <span class="token builtin">range</span><span class="token punctuation">(</span>row<span class="token operator">+</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">:</span>
            Plm<span class="token punctuation">[</span>row<span class="token punctuation">]</span><span class="token punctuation">.</span>append<span class="token punctuation">(</span><span class="token number">0</span><span class="token punctuation">)</span>
    Plm<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span> <span class="token operator">=</span> <span class="token number">1</span>
    Plm<span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span> <span class="token operator">=</span> math<span class="token punctuation">.</span>sqrt<span class="token punctuation">(</span><span class="token number">3</span><span class="token punctuation">)</span> <span class="token operator">*</span> math<span class="token punctuation">.</span>cos<span class="token punctuation">(</span>sita_rad<span class="token punctuation">)</span>
    Plm<span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">=</span> math<span class="token punctuation">.</span>sqrt<span class="token punctuation">(</span><span class="token number">3</span><span class="token punctuation">)</span> <span class="token operator">*</span> math<span class="token punctuation">.</span>sin<span class="token punctuation">(</span>sita_rad<span class="token punctuation">)</span>
    <span class="token keyword keyword-for">for</span> row <span class="token keyword keyword-in">in</span> <span class="token builtin">range</span><span class="token punctuation">(</span><span class="token number">2</span><span class="token punctuation">,</span> n<span class="token operator">+</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">:</span>
        coefficient_1 <span class="token operator">=</span> math<span class="token punctuation">.</span>sqrt<span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token number">2</span> <span class="token operator">*</span> row <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">)</span><span class="token operator">/</span><span class="token punctuation">(</span><span class="token number">2</span> <span class="token operator">*</span> row<span class="token punctuation">)</span><span class="token punctuation">)</span>
        coefficient_2 <span class="token operator">=</span> math<span class="token punctuation">.</span>sqrt<span class="token punctuation">(</span><span class="token number">2</span> <span class="token operator">*</span> row <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">)</span>
        Plm<span class="token punctuation">[</span>row<span class="token punctuation">]</span><span class="token punctuation">[</span>row<span class="token punctuation">]</span> <span class="token operator">=</span> coefficient_1 <span class="token operator">*</span> math<span class="token punctuation">.</span>sin<span class="token punctuation">(</span>sita_rad<span class="token punctuation">)</span> <span class="token operator">*</span> Plm<span class="token punctuation">[</span>row<span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">[</span>row<span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">]</span>
        Plm<span class="token punctuation">[</span>row<span class="token punctuation">]</span><span class="token punctuation">[</span>row<span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">=</span> coefficient_2 <span class="token operator">*</span> math<span class="token punctuation">.</span>cos<span class="token punctuation">(</span>sita_rad<span class="token punctuation">)</span> <span class="token operator">*</span> Plm<span class="token punctuation">[</span>row<span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">[</span>row<span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">]</span>
    <span class="token keyword keyword-for">for</span> row <span class="token keyword keyword-in">in</span> <span class="token builtin">range</span><span class="token punctuation">(</span><span class="token number">2</span><span class="token punctuation">,</span> n<span class="token operator">+</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">:</span>
        <span class="token keyword keyword-for">for</span> clumn <span class="token keyword keyword-in">in</span> <span class="token builtin">range</span><span class="token punctuation">(</span><span class="token number">0</span><span class="token punctuation">,</span> row<span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">:</span>
            coefficient_3 <span class="token operator">=</span> math<span class="token punctuation">.</span>sqrt<span class="token punctuation">(</span><span class="token punctuation">(</span><span class="token number">2</span> <span class="token operator">*</span> row <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token punctuation">(</span>row <span class="token operator">-</span> clumn<span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token punctuation">(</span>row <span class="token operator">+</span> clumn<span class="token punctuation">)</span><span class="token punctuation">)</span>
            coefficient_4 <span class="token operator">=</span> math<span class="token punctuation">.</span>sqrt<span class="token punctuation">(</span><span class="token number">2</span> <span class="token operator">*</span> row <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">)</span>
            coefficient_5 <span class="token operator">=</span> math<span class="token punctuation">.</span>sqrt<span class="token punctuation">(</span><span class="token punctuation">(</span>row <span class="token operator">-</span> clumn <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">)</span> <span class="token operator">*</span> <span class="token punctuation">(</span>row <span class="token operator">+</span> clumn <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">)</span> 
            <span class="token operator">/</span> <span class="token punctuation">(</span><span class="token number">2</span> <span class="token operator">*</span> row <span class="token operator">-</span> <span class="token number">3</span><span class="token punctuation">)</span><span class="token punctuation">)</span>
            Plm<span class="token punctuation">[</span>row<span class="token punctuation">]</span><span class="token punctuation">[</span>clumn<span class="token punctuation">]</span> <span class="token operator">=</span> coefficient_3 <span class="token operator">*</span> <span class="token punctuation">(</span>coefficient_4 <span class="token operator">*</span> math<span class="token punctuation">.</span>cos<span class="token punctuation">(</span>sita_rad<span class="token punctuation">)</span> <span class="token operator">*</span> Plm<span class="token punctuation">[</span>row <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span>
            <span class="token punctuation">[</span>clumn<span class="token punctuation">]</span> <span class="token operator">-</span> coefficient_5 <span class="token operator">*</span> Plm<span class="token punctuation">[</span>row <span class="token operator">-</span> <span class="token number">2</span><span class="token punctuation">]</span><span class="token punctuation">[</span>clumn<span class="token punctuation">]</span><span class="token punctuation">)</span>
    <span class="token keyword keyword-return">return</span> Plm

</code></pre>
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